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Logarithmic Differentiation

The solution ...

Question

The solution of the differential equation $y d x + (x + x^{2} y) d y = 0$ is:

- \frac{1}{x y} = c

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- \frac{1}{x y} +

logy

= c

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\frac{1}{x y} + l o g y = c

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l o g y = c x

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Solution

The correct option is A $- \frac{1}{x y} +$ logy $= c$
Given, $y d x + (x + x^{2} y) d y = 0$
$\Rightarrow y d x + x d y = - x^{2} y d y$
$\Rightarrow \frac{y d x + x d y}{x^{2} y^{2}} = - \frac{1}{y} d y$
$\Rightarrow d (- \frac{1}{x y}) = - \frac{1}{y} d y$
Integrating both sides, we get
$- \frac{1}{x y} = - log y + c$
$\Rightarrow - \frac{1}{x y} + log y = c$

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